2010 AIME I Problem 4

Jackie and Phil have two fair coins and a third coin that comes up heads with probability \frac47. Jackie flips the three coins, and then Phil flips the three coins. Let \frac {m}{n} be the probability that Jackie gets the same number of heads as Phil, where m and n are relatively prime positive integers. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.